where \(N_{\pi }\)  are proton single particle orbits, and \(N_{\nu }\)  are neutron single particle orbits.
After encoding quantum states on the qubits, we use the Jordan-Wigner(J-W) transformation [1]} to map the creation and annihilation operators into the Pauli operators as
\(\begin{aligned}a_{k}^{\dagger } &= c_{k}|1\rangle _{k}\langle 0| \\&= \frac{1}{2}Z_{1}\otimes Z_{2}\otimes ...Z_{k-1}\otimes (X_{k}-iY_{k}) \otimes I_{k+1} \otimes ... \\a_{k} &= c_{k}|0\rangle _{k}\langle 1| \\&= \frac{1}{2}Z_{1}\otimes Z_{2}\otimes ...Z_{k-1}\otimes (X_{k}+iY_{k}) \otimes I_{k+1} \otimes ...\end{aligned}\) 
